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Analysis of Neutral Stochastic Fractional Differential Equations Involving Riemann–Liouville Fractional Derivative with Retarded and Advanced Arguments

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Abstract

This typescript explores the existence and uniqueness of solutions for Riemann–Liouville neutral stochastic fractional differential equations with both retarded and advanced arguments. In order to establish the existence of solution, we applied the M\(\acute{o}\)ach condition, while for uniqueness, we utilized the Banach contraction principle. Furthermore, we studied the Hyers–Ulam stability of these equations. To reinforce our findings, we provide an example at the conclusion of the manuscript.

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References

  1. Ajeet, S., Anurag, S., Vijayakumar, V., Udhayakumar, R.: Asymptotic stability of fractional order \((1,2]\) stochastic delay differential equations in Banach spaces. Chaos, Solitons Fractals 150, 111095 (2021)

    Article  MathSciNet  Google Scholar 

  2. Agarwal, R.P., Ntouyas, S.K., Ahmad, B., Alzahrani, A.K.: Hadamard-type fractional functional differential equations and inclusions with retarded and advanced arguments. Adv. Differ. Equ. 1, 1–15 (2016)

    MathSciNet  Google Scholar 

  3. Ahmed, M., Zada, A., Ahmed, J., Mohamed, A.: Analysis of stochastic weighted impulsive neutral \(\psi -\)Hilfer integro fractional differential system with delay. Math. Probl. Eng. (2022). https://doi.org/10.1155/2022/1490583

    Article  Google Scholar 

  4. Balachandran, K., Park, J.Y.: Controllability of fractional integrodifferential systems in Banach spaces. Nonlinear Anal. Hybrid Syst. 3, 363–367 (2009)

    Article  MathSciNet  Google Scholar 

  5. Bellman, R.: The stability of solutions of linear differential equations. Duke Math. J. 10(4), 643–647 (1943)

    Article  MathSciNet  Google Scholar 

  6. Cui, J., Yan, L.: Existence result for fractional neutral stochastic integro-differential equations with infinite delay. J. Phys. A Math. Theor. 44(33), 335201 (2011)

    Article  MathSciNet  Google Scholar 

  7. Diethelm, K.: The analysis of fractional differential equations. Lecture Notes in Mathematics, (2010)

  8. Deinz, H.: On the behaviour of measure of noncompactness with respect to differentiation and integration of vector-valued functions. Nonlinear Anal. TMA 7, 1351–1371 (1983)

    Article  MathSciNet  Google Scholar 

  9. Deng, S., Shu, X., Mao, J.: Existence and exponential stability for impulsive neutral stochastic functional differential equations driven by fBm with noncompact semigroup via Mönch fixed point. J. Math. Anal. App. 467, 398–420 (2018)

    Article  Google Scholar 

  10. Euler, L.: Institutiones Calculi Differentialis Cum Eius Usu in Analysi Finitorum ac Doctrina Serierum, Academiae scientiarum Imperialis Petropolitanae. (1748)

  11. Granas, A., Dugundji, J.: Fixed point theory. Springer-Verlag, New York (2003)

    Book  Google Scholar 

  12. Gambo, Y., Jarad, F., Baleanu, D., Abdeljawad, T.: On Caputo modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 10, 2014 (2014)

    MathSciNet  Google Scholar 

  13. Hafiz, F.M.: The fractional calculus for some stochastic processes. Stoch. Anal. Appl. 22(2), 507–523 (2004)

    Article  MathSciNet  Google Scholar 

  14. Harisa, S.A., Ravichandran, C., Nisar, K.S., Faried, N., Morsy, A.: New exploration of operators of fractional neutral integro-differential equations in Banach spaces through the application of the topological degree concept. AIMS Math. 7(9), 15741–15758 (2022)

    Article  MathSciNet  Google Scholar 

  15. Hernandez, E., Fernandes, D., Zada, A.: Local and global existence and uniqueness of solution for abstract differential equations with state-dependent argument. Proc. Edinb. Math. Soc. 26, 1–41 (2023)

    MathSciNet  Google Scholar 

  16. Hu, W., Zhu, Q., Karimi, H.R.: Some improved Razumikhin stability criteria for impulsive stochastic delay differential systems. IEEE Trans. Automat. Contr. 64, 5207–5213 (2019)

    Article  MathSciNet  Google Scholar 

  17. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, p. 204. Elsevier, Amsterdam (2006)

    Google Scholar 

  18. Lakshmikantham, V., Leela, S., Devi, J.V.: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge (2009)

    Google Scholar 

  19. Leibniz, G.W.: Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illi calculi genus, Acta eruditorum, 3, 467-473, (1684)

  20. Maji, C., Al Basir, F., Mukherjee, D., Nisar, K.S., Ravichandran, C.: COVID-19 propagation and the usefulness of awareness-based control measures: a mathematical model with delay. AIMS Math. 7(7), 12091–12105 (2022)

    Article  MathSciNet  Google Scholar 

  21. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. John Wiley, USA (1993)

    Google Scholar 

  22. Nisar, K.S., Jagatheeshwari, R., Ravichandran, C., Veeresha, P.: An effective analytical method for fractional Brusselator reaction-diffusion system. Math. Methods Appl. Sci. (2023). https://doi.org/10.1002/mma.9589

    Article  MathSciNet  Google Scholar 

  23. Podlubny, I.: Fractional Differential Equations. Academic Press, USA (1999)

    Google Scholar 

  24. Sun, H., Zhang, Y., Baleanu, D., Chen, W., Chen, Y.: A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simulat. 64, 213–231 (2018)

    Article  Google Scholar 

  25. Sousa, J.V.D.C., Capelas de Oliveira, E.: On the \(\psi \)-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 60, 72–91 (2018)

    Article  MathSciNet  Google Scholar 

  26. Sousa, J.V.D.C., Capelas de Oliveira, E.: Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation. Appl. Math. Letter. 81, 50–56 (2018)

    Article  MathSciNet  Google Scholar 

  27. Stirzaker, D., Grimmett, G.: Stochastic Processes and Their Applications. Springer, UK (2001)

    Google Scholar 

  28. Vijayaraj, V., Ravichandran, C., Nisar, K.S., Valliammal, N., Logeswari, K., Albalawi, Wedad, Abdel-Aty, A.: An outlook on the controllability of non-instantaneous impulsive neutral fractional nonlocal systems via Atangana-Baleanu-Caputo derivative. Arab. J. Basic Appl. Sci. 30(1), 440–451 (2023)

    Article  Google Scholar 

  29. Wang, B., Zhu, Q.: Stability analysis of semi-Markov switched stochastic systems. Automatica 94, 72–80 (2018)

    Article  MathSciNet  Google Scholar 

  30. Wang, H., Zhu, Q.: Global stabilization of a class of stochastic nonlinear time-delay systems with SISS inverse dynamics. IEEE Trans. Automat. Contr. 65, 4448–4455 (2020)

    Article  MathSciNet  Google Scholar 

  31. Xie, W., Zhu, Q.: Self-triggered state-feedback control for stochastic nonlinear systems with Markovian switching. IEEE Trans. Syst. Man Cybern. Syst. 50, 3200–3209 (2020)

    Article  Google Scholar 

  32. Yan, B.: Boundary value problems on the half-line with impulses and infinite delay. J. Math. Anal. Appl. 259, 94–114 (2001)

    Article  MathSciNet  Google Scholar 

  33. Zada, A., Ali, W., Farina, S.: Hyers-Ulam stability of nonlinear differential equations with fractional integrable impulses. Math. Methods Appl. Sci. 40(15), 5502–5514 (2017)

    Article  MathSciNet  Google Scholar 

  34. Zada, A., Ali, W., Park, C.: Ulam’s type stability of higher order nonlinear delay differential equations via integral inequality of Gronwal Bellman, Bihari’s type. Appl. Math. Comput. 350, 60–65 (2019)

    MathSciNet  Google Scholar 

  35. Zada, A., Ali, S.: Stability analysis of multi-point boundary value problem for sequential fractional differential equations with non-instantaneous impulses. Int. J. Nonlinear Sci. Numer. Simul. 19(7), 763–774 (2018)

    Article  MathSciNet  Google Scholar 

  36. Zada, A., Shaleena, S., Li, T.: Stability analysis of higher order nonlinear differential equations in \(\beta \)-normed spaces. Math. Methods Appl. Sci. 42, 1151–1166 (2019)

    Article  MathSciNet  Google Scholar 

  37. Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014)

    Book  Google Scholar 

  38. Zhu, Q.: Stability analysis of stochastic delay differential equations with Lévy noise. Syst. Control. Lett. 118, 62–68 (2018)

    Article  Google Scholar 

  39. Zhu, Q., Wang, H.: Output feedback stabilization of stochastic feedforward systems with unknown control coefficients and unknown output function. Automatica 87, 165–175 (2018)

    Article  MathSciNet  Google Scholar 

  40. Zine, H., Torres, D.F.M.: A stochastic fractional calculus with applications to variational principles. Fractal Fract. 4, 38 (2020)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan

    Shahid Saifullah, Sumbel Shahid & Akbar Zada

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  1. Shahid Saifullah
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Saifullah, S., Shahid, S. & Zada, A. Analysis of Neutral Stochastic Fractional Differential Equations Involving Riemann–Liouville Fractional Derivative with Retarded and Advanced Arguments. Qual. Theory Dyn. Syst. 23, 39 (2024). https://doi.org/10.1007/s12346-023-00894-w

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